Related papers: Stable rank one matrix completion is solved by two…
We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers.…
This paper focuses on recovering an underlying matrix from its noisy partial entries, a problem commonly known as matrix completion. We delve into the investigation of a non-convex regularization, referred to as transformed $L_1$ (TL1),…
In this paper, we propose a low-rank coordinate descent approach to structured semidefinite programming with diagonal constraints. The approach, which we call the Mixing method, is extremely simple to implement, has no free parameters, and…
Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This…
The rank minimization problem is to find the lowest-rank matrix in a given set. Nuclear norm minimization has been proposed as an convex relaxation of rank minimization. Recht, Fazel, and Parrilo have shown that nuclear norm minimization…
Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex…
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy…
We have recently presented a method to solve an overdetermined linear system of equations with multiple right hand side vectors, where the unknown matrix is to be symmetric and positive definite. The coefficient and the right hand side…
This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle…
Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the…
In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to…
The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system…
This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second…
The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This…
The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices.…
We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the…
Given polynomials f(x), g_i(x), h_j(x), we study how to minimize f on the semialgebraic set S = { x \in R^n: h_1(x)=...=h_{m_1}(x) =0, g_1(x) >= 0, ..., g_{m_2}(x) >= 0}. Let f_{min} be the minimum of f on S. Suppose S is nonsingular and…
We consider polynomial optimization problems on Cartesian products of basic compact semialgebraic sets. The solution of such problems can be approximated as closely as desired by hierarchies of semidefinite programming relaxations, based on…
Recht, Fazel, and Parrilo provided an analogy between rank minimization and $\ell_0$-norm minimization. Subject to the rank-restricted isometry property, nuclear norm minimization is a guaranteed algorithm for rank minimization. The…
This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions.…